Field of the Invention
The present invention relates to a control system, a positioning system, a lithographic apparatus and a device manufacturing method.
Description of the Related Art
A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In such a case, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g. including part of, one, or several dies) on a substrate (e.g. a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Conventional lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at once, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
In the known lithographic apparatus a substrate support is provided to support a substrate during transfer of a pattern of the patterning device onto the substrate. A control system is provided to accurately control the position of the substrate support. In a known embodiment of this control system a feedforward is provided. The performance of the control system is highly dependent on the accuracy of the feedforward.
Finite impulse response (FIR) filters are used to improve performance of the feedforward. FIR filters use time shifted operations in order to construct a feedforward signal, i.e. the output of the feedforward is based on the set-point signal of the present time sample and one or more previous time samples.
An example of FIR filters applied in a multi-input-multi output (MIMO) position control system can be found in the publication “Data-Based Feed-Forward control in MIMO Motion Systems” by Mark Baggen, Marcel Heertjes and Ramidin Kamidi, 2008 American Control Conference, Seattle, Jun. 11-13 2008, the contents of which are herein incorporated in its entirety by reference. In this publication, it is proposed to optimize the coefficients of a set of FIR filters on the basis of a quadratic objective function related to a performance-relevant time-frame of the servo error signal using a Gauss-Newton method.
In this method the linear feedforward techniques are used to minimize the error in an iterative way, whereby parameters are perturbed and set such that the error is minimal. This method is also referred to as the machine-in-the-loop procedure.
Although the feed-forward device incorporating FIR filters including coefficients optimized by a Gauss-Newton method may substantially improve the performance of the control system, there is a continuous need to improve the performance of the control system.
In particular, when the dynamic system to be controlled, for example a substrate support, comprises a non-linear characteristic, this non-linear characteristic may have a substantial negative effect on the performance of the control system. In a FIR filter, the non-linearity will have to be mapped during optimization to the linear feedforward coefficients of the FIR filters.
To still meet the desired performance level, this may result in the need of substantial larger FIR filters, and as a consequence, a substantial larger number of feedforward coefficients.
This larger number of feedforward coefficients also increases the time required to calibrate the control system, since each of the coefficients has to be optimized. Thereby, a re-optimization may be required whenever a different set-point is applied.